Social Network Analysis: Centrality Measures

Home , Betweenness centrality, Centrality, Closeness centrality

Faculty of Business Administration, University of New Brunswick, NB Canada Fredericton E3B 9Y2

Donglei Du (UNB) Social Network Analysis 1/85 What is centrality? I

Centrality measures address the question: "Who is the most important or central person in this network?" There are many answers to this question, depending on what we mean by importance. According to Scott Adams, the power a person holds in the organization is inversely proportional to the number of keys on his keyring. A janitor has keys to every oce, and no power. The CEO does not need a key: people always open the door for him. There are a vast number of dierent centrality measures that have been proposed over the years.

Donglei Du (UNB) Social Network Analysis 4/85 Degree centrality for undirected graph I

The nodes with higher degree is more central. Let A Rn n be the adjacency matrix of a undirected graph. 2 ⇥ Let k Rn be the degree vector. Let e Rn be the all-one 2 2 vector. Then k = Ae

For comparison purpose, we can standardize the degree by dividing by the maximum possible value n 1. Degree is simply the number of nodes at distance one. Though simple, degree is often a highly eective measure of the inﬂuence or importance of a node: In many social settings people with more connections tend to have more power and more visible.

Donglei Du (UNB) Social Network Analysis 7/85 Closeness centrality for undirected graph

The farness/peripherality of a node v is deﬁned as the sum of its distances to all other nodes The closeness is deﬁned as the inverse of the farness. 1 closeness(v)= Âi=v dvi 6

For comparison purpose, we can standardize the closeness by dividing by the maximum possible value 1/(n 1) If there is no (directed) path between vertex v and i then the total number of vertices is used in the formula instead of the path length. The more central a node is, the lower its total distance to all other nodes. Closeness can be regarded as a measure of how long it will take to spread information from v to all other nodes sequentially.

Donglei Du (UNB) Social Network Analysis 15 / 85 Betweenness centrality

Betweenness centrality quantiﬁes the number of times a node acts as a bridge along the shortest path between two other nodes. It was introduced as a measure for quantifying the control of a human on the communication between other humans in a social network by Linton Freeman. In this conception, vertices that have a high probability to occur on a randomly chosen shortest path between two randomly chosen vertices have a high betweenness.

Donglei Du (UNB) Social Network Analysis 17 / 85